Ductility, snapback, size effect, and redistribution in softening beams or frames

Zdenëk P. Bažant, Gilles Pijaudier-Cabot, Jiaying Pan

Research output: Contribution to journalArticlepeer-review

34 Scopus citations


A layered finite element model with strain-softening material properties, whose applicability to reinforced concrete was corroborated by comparisons with experimental data in the preceding paper, is used in a parametric study aimed at the effect of several factors: Structure size, finite element size, downward slope of strain-softening stress-strain relation, length of the plastic yield plateau before the onset of strain softening (if any), and end-restraint stiffness. To quantify the response, several new response characteristics are introduced: The ductile strengthening factor, characterizing how strain softening reduces the maximum load compared to the plastic limit load; the redistribution ratio, characterizing the degree of bending moment redistribution in comparison to that in plastic limit analysis; the energy safety factor, describing the energy to deform the structure to the peak load; and the ductility factor, characterizing the deflection increase at maximum load relative to the deflection from elastic analysis. The condition of snapback instability, which determines the ductility factor, is derived analytically for an elastically restrained beam. Finally, it is shown that strain-softening segments in beams cannot be modeled as softening hinges except for sufficiently slender beams.

Original languageEnglish (US)
Pages (from-to)2348-2364
Number of pages17
JournalJournal of Structural Engineering (United States)
Issue number12
StatePublished - Dec 1987

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Building and Construction
  • Materials Science(all)
  • Mechanics of Materials
  • Mechanical Engineering


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